LABORATORY SCIENCE

Model of accommodatio

n: Contributions of lensgeometry and mechanical properties to the

development of presbyopiaDominique Van de Sompel, Gary J. Kunkel, PhD, Peter S. Hersh, MD, Alexander J. Smits, PhD

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P

1960

2010 A

ublished

PURPOSE: To determine the relative importance of lens geometry and mechanical properties for themechanics of accommodation and the role of these elements in the causes and potential correctionof presbyopia.

SETTING: Department of Mechanical and Aerospace Engineering, Princeton University, Princeton,New Jersey, USA.

DESIGN: Experimental study.

METHODS: Finite element methods and ray-tracing algorithms were used to model the deformationand optical power of the human crystalline lens during accommodation. The mechanical modeltreats the lens as an axisymmetric object, and the optical model incorporates a gradientrefractive index. Using these models, the accommodation of a broad range of lenses withdifferent geometries and mechanical properties were investigated.

RESULTS: The most significant result was that reshaping the 45-year-old lens to the geometry ofthe 29-year-old lens, while retaining the mechanical properties, restored the former’s accommo-dation amplitude to 72% to 94% of that of the 29-year-old lens, depending on ciliary bodydisplacement. That is, reshaping can add 1.8 to 3.7 diopters of accommodation. A sensitivityanalysis showed that this result was robust over a wide range of mechanical and geometricalproperties.

CONCLUSION: The study results suggest that a significant amount of the loss of accommodation isdue to changes in lens geometry.

Financial Disclosure: No author has a financial or proprietary interest in any material or methodmentioned.

J Cataract Refract Surg 2010; 36:1960–1971 Q 2010 ASCRS and ESCRS

The ability of humans to change their visual focus bet-ween far objects and near objects is governed by thebiomechanical process of lens accommodation. Themechanics of this process are generally agreed to fol-low Helmholtz’s original hypothesis, which suggeststhat ciliary body contraction causes a relaxation inthe stress on zonular fibers, allowing the lens to bulgeto its less-stressed state. In this accommodated state,the equatorial diameter is decreased, the optical axisthickness is increased, and the anterior and posteriorlens surfaces have smaller radii of curvature. Thisdecreased radius of curvature gives the lens a higherrefractive power. In the unaccommodated state, theciliary body is relaxed, the zonular fibers are taut,and the lens is flattened, resulting in a lower refractivepower. While favored by the overwhelming body of

SCRS and ESCRS

by Elsevier Inc.

scientific evidence,1 Helmholtz’s theory is not theonly one proposed. An alternative theory proposedby Schachar et al.2 states that both the lens equatorand zonular tension increase on accommodation.Recent work has provided evidence that the 2 theoriesmay not be mutually exclusive and that lens age maybe the dominant factor in determining which theoryapplies.3 In other words, recent work has demon-strated that the 2 theories of accommodation maycoexist. Although the mechanics of the accommoda-tion process have been well studied1,4 and competingtheories somewhat reconciled, the root cause of thefailure of this mechanism (presbyopia) has not beenagreed on.

Presbyopia is thought to result from geometric andmchanical property changes in the accommodative

0886-3350/$dsee front matter

doi:10.1016/j.jcrs.2010.09.001

1961LABORATORY SCIENCE: CONTRIBUTIONS OF LENS GEOMETRY AND MECHANICAL PROPERTIES TO PRESBYOPIA

system (ie, lens, ciliary body, zonular fibers, and aque-ous and vitreous humors) with age. However, there isno consensus as to the primary cause of presbyopia orthe relative importance of changes in the geometric,mechanical, and optical properties of the various com-ponents of the accommodative system. For instance,presbyopia may result from loss of the contractionability of the ciliary muscle,5 hardening of the lens,6

or thickening of the lens.1 Although other causeshave been suggested, these seem to be the most prob-able based on empirical1 and computational evi-dence.7,8 There are several review articles1,4 of themost likely causes of presbyopia, but we will not re-view them. Instead, in this study, we focus on the re-cent findings of Strenk et al.1, which suggest thatlenticular changes are solely responsible for the devel-opment of presbyopia in humans. (These findings aredistinctly different from findings in some primates, asdiscussed in Strenk et al.1). Strenk et al.'s findings alsoagree with simulation results of Martin et al.,8 who re-port that an applied pressure load on the posterior lenssurface is not essential for the accommodation process.

Various ways to correct presbyopia (restore accom-modation) have been suggested. A recent editorial byCharman9 places these approaches into 3 main cate-gories: those that replace the lens with a man-madebiocompatible material, those that use scleral expan-sion bands or other surgical modalities to expand theciliary ring, and those that use intraocular lenses(IOLs). These categories are consistent with the reviewof current accommodation restoration concepts

Submitted: June 13, 2009.Final revision submitted: April 15, 2010.Accepted: May 13, 2010.

From the Department of Mechanical and Aerospace Engineering(Van de Sompel, Kunkel, Smits), Princeton University, Princeton,and the Institute of Ophthalmology and Visual Science (Hersh),University of Medicine and Dentistry of New Jersey, Newark,New Jersey, USA.

Supported in part by the Princeton Institute for the Science andTechnology of Materials. Dominique Van de Sompel was sup-ported by the Morgan W. McKinzie 1993 Senior Thesis Fund anda Davis United World Colleges Scholarship, Princeton University.Dr. Kunkel was partially funded by Princeton University as aCouncil on Science and Technology Postdoctoral Teaching Fellow.

Gowri Rao, Department of Mechanical and Aerospace Engineering,Princeton University, performed some of the early work in thestudy.

Corresponding author: Dominique Van de Sompel, Department ofMechanical and Aerospace Engineering, Engineering Quad, OldenStreet, Princeton University, Princeton, New Jersey, 08544-5263,USA. E-mail: dominiqu@gmail.com.

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presented by Glasser.4 In agreement with the findingsof Strenk et al.,1 we believe that restoration of the ac-commodative amplitude by surgically modifying theexisting crystalline lens holds enormous potential.While we make no claims about the feasibility of thisapproach, it does provide motivation for trying to bet-ter understand the fundamental physics of accommo-dation. Other researchers such as Gerten et al.10 (aswell as colleagues at Princeton) have begun applyingthis logic and started studying the effects of femtosec-ond laser ablation of lenses with applications towardthe correction of presbyopia. To be successful, the re-searchers will have to know what parameters of thelenticular system to change and how to change them.

We present our findings on the relative importanceof geometric and mechanical properties of the humancrystalline lens in the accommodation process and thepotential correction of presbyopia. We use a finite-element analysis similar to the model of Burd et al.7

to investigate the relationship between the changinggeometric and mechanical properties of the lens withage to understand which components have a greatereffect on the overall cause of presbyopia. The originalmechanical model of Burd et al.7 is used alongwith thegradient refractive index model of Blaker11 to accountfor the change in the index of refractionwithin the lens.We have based our study on Burd et al.'s lens modelbecause the finite element code for simulating itsdeformations during accommodation is freely availa-ble.A We believe that this adds to the reproducibilityof our work. To account for the gradient index ofrefraction, a ray-tracing optical model is used to calcu-late the focal length of the lens and the accommodativeamplitude, which is the difference in the optical poweratmaximum andminimum ciliary body displacement.This ray-tracing algorithm allows us to account for thegradient index of refraction, and it also allows us toinvestigate the effects of inserting a substance witha different refractive index into the lens. (Charman9

presents examples of several circumstances in whichthismight be useful.) Our results show there is a differ-ence in the calculated optical power of a uniform lensand that of a lens with a gradient index of refraction.Ray-tracing is also useful for understanding the factorsthat contribute to any astigmatism that may occur,although this is not studied in detail in this work.The accommodative amplitudes we discuss are valuesfor the lens in isolation. Burd et al.7 cite Bennett,12 whoshows that the accommodative amplitude of the entireeye is 76% of the value for the lens. To critique thesignificance of any restoration of accommodation,we compare our findings with those of Weale,13

who suggests that to be effective, IOLs must beable to increase the optical power of the lens by 3.0to 4.0 diopters (D).

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1962 LABORATORY SCIENCE: CONTRIBUTIONS OF LENS GEOMETRY AND MECHANICAL PROPERTIES TO PRESBYOPIA

In this study, we make use of the lens geometric andmechanical properties given by Burd et al.7 However,Burd et al.14 suggest that modeling assumptions inFisher’s original analysis of his spinning lens test15

may have led to systematic errors in the determinationof the Young modulus of the nucleus and cortex. Theyspecifically remind us that the accuracy of any finite-element calculation ultimately depends on the geo-metric and mechanical properties fed into the model.Therefore, we checked the robustness of our conclu-sions by conducting a sensitivity analysis of the geo-metric and mechanical properties based on a rangeof property values given in the literature. As shownin the results section, our general conclusion that theaccommodative capacity of a lens can be largelyrestored by reshaping it holds over a large range ofgeometric and mechanical properties given in theliterature.

MATERIALS AND METHODS

Mechanical Model

The finite-element analysis package Abaqus 6.5-1 usingaxisymmetric models of the human crystalline lens at differ-ent ages was used to simulate the stress conditions in accom-modation. Each of the lens models was composed of 3distinct regions: the nucleus, the cortex, and the capsule.For eachmodel, it was assumed that the fully accommodatedlens was entirely stress free, providing a convenient refer-ence state from which to start the simulations. The conclu-sions are based on modeling the geometric and mechanicalproperties of the 29-year-old and 45-year-old lens as usedby Burd et al.,7 which were originally derived from clinicaldata collected by Brown.16 As mentioned, a wide range ofgeometric and mechanical properties were also simulatedto evaluate the sensitivity of the conclusions to any errorsin the values used by Burd et al.7 Burd et al.7 report the stiff-ness values of the nucleus and cortex to be En Z 0.5474 �10�3 MPa and Ec Z 3.417 � 10�3 MPa, respectively, for the29-year-old lens and En Z 0.9966 � 10�3 MPa and Ec Z3.980 � 10�3 MPa, respectively, for the 45-year-old lens.However, other studies report stiffness values that fallwithin a range of magnitudes. Abolmaali et al.17 report stiff-nesses for both nucleus and cortex that range from 0.5� 10�4

to 8.0 � 10�4 MPa, and van Alphen and Graebel18 reportstiffnesses that range from 4.4 � 10�3 to 10.9 � 10�3 MPa.Heys et al.6 report a much wider range, suggesting that theYoung modulus of the nucleus (En) ranges from 25.7 �10�6 MPa (20 years) to 23.9 � 10�3 MPa (73 years). The cor-tex’s Young modulus (Ec) was determined to range from48.5 � 10�6 MPa (14 years) to 2.577 � 10�3 MPa (76 years).(The results of Hey et al.6 should be treated with caution.The study was conducted on frozen tissue, and the validityof applying material properties of frozen tissue to fresh tis-sue have been questioned19 because they are not physiologi-cally realistic.) In light of these varying reports, therobustness of our conclusions was assessed by varying theYoung moduli of the nucleus and cortex between the mini-mum values reported by Heys et al.6 and the maximumvalues reported by Burd et al.7 The cortical and nuclear stiff-nesses were increased from their minimum to maximum

J CATARACT REFRACT SURG - V

values in 10 increments. The mechanical properties of allother lens parts were set to those used by Burd et al.7

Next, the anterior and posterior surfaces of the lens weredescribed with separate polynomials using empirical coeffi-cients obtained from Burd et al.7 The curvature of theselenses varied along the surface but was equal to 2.3 and 5.8at the anterior and posterior poles of the 29-year-old lensand to 0.1 and 3.2. at the anterior and posterior poles of the45-year-old lens. The equatorial region of the lens was de-fined using a circular end cap that ensured continuity ofthe slope (but not necessarily of the curvature). Three setsof zonular fibers were assumed to connect with the ciliarymuscle at a single point, located at the same axial coordinateas the equator of the lens, and at the ciliary body radius. Theattachment of the equatorial zonule was assumed to remainstationary on the lens equator with age, and the posterior at-tachments were assumed to be located at the same projecteddistance from the lens equator as the anterior attachments.Following Burd et al.,7 the boundary between the nucleusand the cortex was modeled by circular arcs. The nucleusand the cortex were both contained in the capsule, a thinouter membrane. Uncertainties in the exact geometry of thelens were considered by using various lens thicknesses inthe simulations, as discussed in the results section.

To simulate the deformation of each lens during accom-modation, a displacement d was applied incrementally tothe point of attachment to the ciliary body. The resulting ge-ometries were then computed using the nonlinear solutionalgorithms available in Abaqus. The maximum displace-ments were the same as those used by Burd et al.,7 whichwere determined from an empirical curve fit of data fromStrenk et al.,20 namely

dmaxðmmÞZ0:5129� 0:00525� age ðyearsÞ (1)

which yields maximum displacements of 0.361 and 0.277 mmfor the 29-year-old lens and 45-year-old lens, respectively.

To investigate the effects of lens reshaping (and materialmodification), the geometric and mechanical properties ofthe 45-year-old lens reported by Burd et al.7 were used andthe parameters were individually modified; ie, the geometricand mechanical properties of the 29-year-old lens reportedby Burd et al.7 were used as test cases. This and other inves-tigations are discussed in detail in the results section.

Optical Model

The lens models were equipped with gradient refractiveindex distributions and a ray-tracing algorithm was devel-oped to determine the path of light rays traveling througheach lens model. Experiments on excised cadaver lenseshave shown that the refractive index of the crystalline lens in-creases from the outer to the inner parts. For example, themost common method for measuring refractive indices isto take samples from different regions of the lens and mea-sure their refractive indices in a refractometer. This methodwas used by Palmer and Sivak21 and Jagger22 to measurethe refractive index distributions in various animal lensesand isolated cat lenses, respectively. An alternative methodfor refractive index determination is the interferencemethod,which was used by Nakao et al.23,24 to examine rabbit lensesand excised human lenses. Nakao et al.23 found that the re-fractive index in rabbit lenses is distributed along ellipticalisoindical lines. These lines follow the shape of the rabbitlens, which means that the refractive index is constant alongits outline. Subsequent researchers21,25 incorporated this

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Figure 1. Isoindical contours inside the model of the crystalline lens.The dashed vertical line indicates the border between the anterior andposterior parts of the lens. (Isolines computed using the model pro-vided by Blaker.11) The variables xa, xp, xeq and yeq denote the coor-dinates of the anterior pole, posterior pole, and equator, respectively.The coordinate x’ denotes the axial distance relative to the lensequator.

1963LABORATORY SCIENCE: CONTRIBUTIONS OF LENS GEOMETRY AND MECHANICAL PROPERTIES TO PRESBYOPIA

form of the refractive index distribution into their models ofbovine as well as human lenses. The models yielded goodagreement with experimental ray-path parameters in bovinelenses but provided inaccurate predictions for light rays inhuman lenses. One explanation, proposed by Popiolek-Masajada and Kasprzak,26 was that the assumption of con-centric, isoindical shells following the lens profile was notvalid. Pierscionek27 measured refractive index values of invitro human lenses along the equatorial and sagittal planesusing a fiber-optic probe and found that the refractive indexof the human lens is higher at the lens poles than at the lensequator. The refractive index of the human lens is thus notconstant along the lens outline and does not follow theisoindical shapes in rabbit (or bovine) lenses.

In this study, it was assumed that the refractive index dis-tribution in the human lens is parabolic. This choice of indexdistribution was proposed by Blaker11 and conforms to theobservation that the refractive index distribution is not con-stant along the lens outline. While the parabolic form wasalso supported by Nakao et al.,24 it should be noted thatthe exact form of the refractive index distribution has notbeen widely agreed on. Some studies27,28 suggest a higherpolynomial form, which flattens across the nuclear region.However, as shown in the latter study, substantial differ-ences between individuals can occur and differences in theform of the profiles can possibly be explained by differencesin the ages of the subjects. This matter is beyond the scope ofour study, inwhich the objective was to elucidate the relativeimportance of the shape and themechanical properties of thelens given a particular refractive index distribution. Further-more, in this paper, we show that our conclusions also holdfor optically uniform lenses, suggesting that less drastic dif-ferences in the distributions (eg, 2nd-order versus higherpolynomial profiles) may not change our conclusions.Whether this is truly the case is a topic worthy of furtherinvestigation.

The parabolic form also allowed us to verify the results ofthe ray-tracing program analytically. Moore29 proposed ananalytical method for paraxial ray tracing through polyno-mial refractive index distributions, such as the one used inour lens model. (The analytical verification of our ray tracingalgorithm will not be repeated in this paper.) The parabolicrefractive index distributions were expressed as

nðx;yÞZN0:0 þN0:1xþN0:2x2 þN1:0y2 (2)

where x and y represented the axial and radial coordinates ofthe lens, respectively, both expressed in millimeters. Sincethe index distributions were offset by xeq (Figure 1), wefound it easier to express their definitions as

nðx0;yÞZN0:0 þN0:1x0 þN0:2x02 þN1:0y2 (3)

where x’ Z x � xeq. Furthermore, since the lens was asym-metric about the equatorial plane, it was divided into ante-rior and posterior parts and parabolic index distributionswere fitted to each half. The anterior and posterior indexdistributions were denoted by nant(x’,y) and npost(x’,y),respectively.

To determine the 4 coefficients of each refractive index dis-tribution, 4 boundary conditions were needed. In this model,the benchmark lens indices specified by Blaker11 were used;namely, nZ 1.385 at the anterior pole, nZ 1.387 at the pos-terior pole, nZ 1.406 at the lens center, and nZ 1.375 at thelens equator. These values were assumed to remain constantwith age and are illustrated in Figure 1. The benchmark re-fractive indices were held fixed because the water content

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of the lens seems to remain essentially constant with age30

and the variation of ultrasound velocity in the lens withage is small.31 The boundary conditions for nant(x’,y) andnpost(x’,y) were written as

nant�xa � xeq;0

�Z1:385

nantð0;0ÞZ1:406nant

�0;yeq

�Z1:375

npost�xp � xeq;0

�Z1:387

npostð0;0ÞZ1:406npost

�0;yeq

�Z1:375

The fourth boundary condition was provided by the require-ment that the index distribution reaches a maximum at thecenter of the lens. Following Burd et al.,7 the refractive indexof the surrounding medium was assumed to be 1.336. Thecorresponding distribution coefficients are shown in Table 1.

Note that the same procedure was used to fit gradient in-dex distributions to the deformed profiles. This was tanta-mount to linearly scaling the gradient distribution of thefully accommodated lens in accordance with the motion ofthe 4 benchmark points: the lens center, the 2 lens poles,and the lens equator. This procedure is inaccurate near theequator of the lens, where the deformations are more severethan at the center of the lens. However, our interest was thepath of rays in the polar regions of the lens, where the defor-mations were closest to uniform, since only the polar regions

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Table 1. Index coefficients for the fully accommodated 29-year-old crystalline lens.

Distribution Coefficient Anterior Section Posterior Section

N0,0 1.4060 1.4060N0,1 0 0N0,2 �0.0090 �0.0028N1,0 �0.0017 �0.0017

1964 LABORATORY SCIENCE: CONTRIBUTIONS OF LENS GEOMETRY AND MECHANICAL PROPERTIES TO PRESBYOPIA

are involved in the actual optical pathway of the eye. In theseregions, the gradient index distributions were fitted as be-fore, with minimum resulting error. It was also assumedthat the 45-year-old modified lenses possess a continuous in-dex distribution much like that of the original 29-year-oldand 45-year-old lenses. In reality, the exact distribution afterany potential surgical modification is unknown. As shown inthe results section, however, modifying the geometry of thelens has a much greater affect on the accommodative ampli-tude than modifying the index of refraction. Therefore, themore complex investigation of the modification of the gradi-ent index of refraction is left for future work.

To enable the application of Snell law, the continuousindex distribution was considered as a series of discreteinterfaces between media of infinitesimally different refrac-tive indices. These interfaces corresponded to the isoindicallines of the refractive index distribution. The local normalto an isoindical line at any point (x’, y) was given by the localgradient vector at that point:

VnZvnvx0

iþ vnvy

jZ ðN0;1 þ 2N0;2x0Þiþ ð2N1;0yÞj (4)

where i and j are the unit vectors defining the x- and y-direc-tions, respectively. Equation 4 determines the direction nor-mal to the local isoline at any point in the index distribution,and the newdirection of the light ray at the beginning of eachstep length, much like an explicit integration scheme. Theray-tracing algorithm used a variable step length. Whentraveling inside the gradient index distribution, the steplength was set to 0.01 mm; when traveling outside the lens,it was set to 1.0 mm. This ensured that the resolution of theray path was high only where necessary and decreased theruntime of the algorithm by a factor of 5 compared with us-ing a fixed step length of 0.01 mm. The satisfactory accuracyof these step lengthswas ascertained by comparisonwith theresults of analytical analyses following Moore.29 The ray-tracing algorithms were also tested by comparing the opticalpower calculations with those from the thick lens equationfor the uniform lenses. In all instances, the ray-tracing andthick-lens equations agreed to within 0.6% for the uniformlenses. The ray-tracing results were slightly different becausethey accounted for the effects of spherical aberration on theoverall optical power of the lens.

Experimental design

The experiments were divided into 3 major parts, referredto as parts A, B, and C. In part A, the lens model as describedabove was used to compare the effect of (1) changing the ge-ometry of the lens and (2) changing the lens’ mechanicalproperties on accommodative amplitude. The first changeinvolved reshaping the 45-year-old lens to the geometry of

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the 29-year-old lens, noted as 45 GM (geometry modified).It is not knownwhat dmax of such a lenswould be, but it is ex-pected to lie somewhere between that of the original 29- and45-year-old lenses. Results for both displacements are given.The second change involved replacing the mechanical prop-erties of the 45-year-old lens with those of the 29-year-oldlens, denoted as 45 MM (material modified). The objectiveof part A was to assess which modification, 45 GM or45 MM, achieved the greatest restoration in accommodativeamplitude.

In part B, the sensitivity of the results in part A to varia-tions in the specific mechanical and geometrical propertiesused in the lens models was assessed. This is important sincethe literature about the mechanical properties and geometri-cal dimensions of the human lens contains a range of possi-ble values. The sensitivity analysis in the mechanicalproperties was carried out for the 45-year-old lens and the45 GM lens. More precisely, a range of different stiffnessesfor the nucleus and cortex of each lens was tried based on re-ported values in the literature (see Discussion section). Forthe geometry sensitivity analysis, the most thorough ap-proach would be to consider the distribution of geometriesof prepresbyopic and postpresbyopic lenses in the generalpopulation. Unfortunately, to our knowledge, no detailedaccount of the natural variation in lens geometries acrossthe human population has been published. Although severalstudies in the literature report various types of geometricmeasurements of lenses, using slitlamp photography ormagnetic resonance imaging, none has enough detail toencompass the complete geometrical distribution of prepres-byopic and postpresbyopic lenses.

Therefore, the geometry sensitivity studywas approachedin 2 ways. First, for an initial attempt at characterizing thedistribution of prepresbyopic and postpresbyopic lenses,the minimum and maximum polar thickness values foreach lens model from Strenk et al.20 were used and thechange in accommodative amplitude due to lens reshapingwas recalculated. Second, the effects of changes in the lensgeometry were studied by continuously scaling the geome-tries given by Burd et al.7 More precisely, 2morphing studieswere conducted in which series of target geometries weregenerated and the improvement in accommodative ampli-tude over that of the original 45-year-old lens was calculated.A secondary objective was to determine which target shapewould yield the best improvement in accommodative ampli-tude over that of the 45-year-old lens geometry. In the firstmorphing study, the reference geometry of the 45-year-oldlens was gradually scaled down until its dimensions weresimilar to those of the 29-year-old lens. At each step alongthe way, the accommodative amplitude of the lens thus cre-ated was computed; ie, the downscaling was achieved by in-dependently downscaling the coordinates xa, xp, and yeq ofthe 45-year-old lens. The final scaling factors were arbitrarilychosen so at the end of the morphing process, the axial coor-dinates of the anterior and posterior poles of the reshaped45-year-old lens were scaled to 80% of the correspondingcoordinates in the 29-year-old lens. The radial coordinates,in turn, were scaled in such a way that by the end of themorphing process, the radial coordinate of the lens equatorcoincided with that of the lens equator of the 29-year-oldlens. All optical powers were calculated for gradient refrac-tivity index (GRIN) distributions using the ray-tracingalgorithm. The second morphing study began with the29-year-old geometry and then a varying scaling factorwas applied to the axial coordinates of that lens. In this

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manner, the lens was continuously thinned and thickened,with minimum and maximum scale factors of 75% and125%, respectively.

In part C, the effect of changing the optical properties ofthe lens was examined. Ho et al.32 suggest that replacingexisting lens material with material of a uniform index ofrefraction can be used in the correction of presbyopia. Totest this hypothesis, the nucleus and cortex of each of ourbaseline lens models was replaced with a material of thesame mechanical properties but a different, uniform refrac-tive index. The nucleus’ index of refraction was then variedto examine the effects on the optical power of the lens.

Table 2. Accommodative amplitude for the modified lenses.

Lensdmax

(mm)UniformAA (D)

GRINAA (D)

% of 29GRIN

Above 45GRIN (D)

29-year-old 0.361 9.21 8.49 d 4.2045-year-old 0.277 3.13 4.29 51% d

45 GM1 0.361 8.18 7.99 94% 3.7045 GM2 0.277 6.32 6.10 72% 1.8145 MM 0.277 3.98 5.12 60% 0.83

AA Z accommodative amplitude; GRIN Z gradient refractive index

Figure 2. Optical power versus ciliary body displacement for geo-metric and mechanical lens modifications.

RESULTS

The results of part A are summarized in Figure 2,which shows optical power as a function of ciliarybody displacement for a number of lens models. Thesemodels include the original 29-year-old and 45-year-old lenses equipped with both uniform and gradientrefractive index distributions, as well as the 45 GMand 45 MM modifications. The modified versions ofthe 45-year-old lens model are shown in dotted blacklines. Notably, the 29-year-old and 45-year-old uni-form lenses in Figure 2 agree with the results givenin Burd et al., as expected. Numerical values for the ac-commodative amplitudes of all lenses shown inFigure 2 are given in Table 2.

The results of the mechanical property sensitivityanalysis of part B are shown in Tables 3 to 5. Table 3shows the accommodative amplitudes of the 45-year-old lens geometry with various nuclear and corticalstiffnesses, a gradient refractive index, and amaximumdisplacement of 0.277 mm. Tables 4 and 5 show the in-crease in accommodative amplitude obtained by re-shaping the 45-year-old lens models given in Table 3to the 45 GM model at both values for dmax, namely,0.361 mm (GM1) and 0.277 mm (GM2). Figure 3 illus-trates some of the data in Table 3 by showing opticalpower curves for the 45-year-old GRIN lens at 2 corti-cal and a larger range of nuclear stiffnesses.

Table 6 shows the polar thickness values used forthe source and target geometries in the first part ofthe geometry sensitivity study. Table 7 shows theincrease in accommodative amplitude due to lensreshaping corresponding to the polar thickness valuesin Table 6 for the 2 values of dmax considered in thispaper.

Figure 4 is a representative schematic of themodeledsystem in part C. Figures 5 and 6 show the results ofpart C, in which the nucleus and cortex were replacedwith a new material of a uniform index of refraction.Figure 5 shows the optical power of the eye as a func-tion of ciliary body displacement and nucleus indexof refraction, where the material that replaces thenucleus is denoted as “infill”. Figure 6 shows the

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maximum optical powers and accommodative ampli-tudes of these lenses as a function of infill index ofrefraction.

DISCUSSION

Effect of Changing the Geometric and MechanicalProperties

The overall trends in Figure 2 are similar for all lensmodelsdie, optical power decreased with increasingddalthough they varied in maximum optical powerand accommodative amplitude. In the 29-year-oldlens, the main effect of incorporating the gradient re-fractive index was to shift the optical powers to lowervalues. The opposite trend was observed in the45-year-old lens. This is explained by the action of 2opposing tendencies. First, as illustrated in Figure 1,the refractive index at the edges of the GRIN lensesused in this study is less than that of the correspondinguniform lenses (which have a refractive index of 1.42).Hence, the refraction at the lens edges is less strong forthe GRIN lenses than for the uniform lenses, tending to

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Table 3. Accommodative amplitude (in diopters) of the 45-year-old lens geometry with a gradient refractive index for a maximum dis-placement of 0.277 mm.

En / 0.088 0.189 0.29 0.391 0.492 0.593 0.694 0.795 0.896 0.997Ec Y

0.121 3.34* 2.70* 2.00* 2.77* 2.12* 1.78* 1.40* 0.88* 0.48* 1.49*0.550 4.19* 3.79* 3.38* 3.17* 2.89* 2.74* 2.45* 2.44* 2.26* 2.06*0.979 4.93 4.23 3.92 3.64 3.45 3.24 3.05 2.90 2.64 2.631.407 5.30 4.76 4.32 4.12 3.81 3.84 3.54 3.21 3.37 3.281.836 5.78 4.94 4.52 4.27 3.83 3.96 3.93 3.58 3.37 3.372.265 6.55 5.39 4.79 4.70 4.25 4.21 3.90 3.88 3.77 3.592.694 6.39 5.61 5.21 4.87 4.56† 4.54† 4.26† 4.09† 4.15† 3.83†3.122 6.75 5.66 5.31 5.06 4.78† 4.49† 4.39† 4.36† 4.26† 4.04†3.551 6.86 6.07 5.51 5.48 5.00† 4.88† 4.42† 4.38† 4.35† 4.26†3.980 6.86 6.05 5.62 5.66 5.10† 4.81† 4.62† 4.76† 4.40† 4.29†

Nuclear and cortical stiffnesses (En and Ec) are in units of 10�3 MPANote that within a given data source, the 29-year values are in the upper left and the 45-year values in the lower right and increasing both moduli with agegenerally follows the diagonal of that data*Range that represents the data presented by Heys et al.6 for the 29-year-old lens†Range that represents the data presented by Burd et al.7 for the 45-year-old lens

1966 LABORATORY SCIENCE: CONTRIBUTIONS OF LENS GEOMETRY AND MECHANICAL PROPERTIES TO PRESBYOPIA

lower the optical power of the GRIN lenses. Second,unlike in the uniform lenses, in the GRIN lenses, thelight continues to bend toward the lens’ central axes,tending to increase the overall optical power of thelens. Which of the 2 opposing effects dominatesdepends on the lens thickness; the thicker the lens,the stronger the increase in optical power due to grad-ual internal light refraction. In our study, transitioningfrom uniform to gradient refractive index distributioncauses the thinner 29-year-old lens to decrease in opti-cal power and the thicker 45-year-old lens to increasein optical power. However, whether a GRIN lens hasa higher or lower optical power than its uniform coun-terpart is of little importance to this study. Instead, wewere interested in the relative importance of changes

Table 4. Increase in the accommodative amplitude (in diopters) when thing a maximum ciliary body displacement of 0.361 mm (GM1).

En / 0.088 0.189 0.29 0.391 0.492Ec Y

0.121 5.56 5.71 6.16 4.89 4.950.550 5.16 4.58 4.59 4.30 4.390.979 4.24 4.06 4.35 4.41 4.181.407 4.36 4.11 4.17 3.97 4.041.836 4.05 4.15 4.09 3.90 4.082.265 3.37 3.72 3.96 3.51 3.712.694 3.74 3.88 3.58 3.47 3.683.122 3.61 3.86 3.91 3.65 3.653.551 3.92 3.39 3.38 3.18 3.473.980 3.92 3.60 3.56 3.09 3.62

The nuclear and cortical stiffnesses (En and Ec) used are listed in units of 10�3 MP

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in mechanical and geometric properties to the accom-modative amplitude of a lens with a given refractiveindex distribution.

The 45 GM achieved a drastic improvement in opti-cal power at low ciliary body displacements andachieved a higher accommodative amplitude thanthe original 45-year-old lens whether the lower or up-per limit for dmax was used. The 45 MM, on the otherhand, was less successful: The optical power remainedthe same at low displacements, and the overall accom-modation amplitude increased only slightly, changingless than 1.0 D at the original dmax of the 45-year-oldlens.

As seen in Table 2, depending on the maximumciliary body displacement dmax used, reshaping the

e 45-year-old lens is reshaped to the 29-year-old lens geometry us-

0.593 0.694 0.795 0.896 0.997

5.16 5.32 5.70 5.68 4.694.11 4.24 4.19 4.11 4.414.11 4.24 4.02 4.23 4.103.75 4.03 4.26 3.62 3.733.94 3.78 4.09 3.81 3.863.62 3.67 3.55 3.73 3.853.51 3.69 3.72 3.32 3.493.69 3.56 3.36 3.46 3.313.32 3.77 3.76 3.54 3.383.44 3.58 3.26 3.43 3.70

A

OL 36, NOVEMBER 2010

Table 5. Increase in the accommodative amplitude (in diopters) when the 45-year-old lens is reshaped to the 29-year-old lens geometry us-ing a maximum ciliary body displacement of 0.277 mm (GM2).

En / 0.088 0.189 0.29 0.391 0.492 0.593 0.694 0.795 0.896 0.997Ec Y

0.121 3.51 3.65 4.00 3.05 3.17 3.41 3.49 4.01 4.18 2.970.550 3.04 2.83 2.97 2.76 2.81 2.51 2.65 2.63 2.64 3.020.979 2.29 2.22 2.62 2.63 2.40 2.41 2.57 2.54 2.63 2.421.407 2.49 2.29 2.33 2.27 2.18 2.21 2.24 2.54 2.00 2.281.836 2.20 2.29 2.33 2.15 2.36 2.22 1.96 2.38 2.22 2.212.265 1.56 1.97 2.22 1.82 1.98 1.94 2.03 1.92 2.12 2.312.694 1.78 2.03 1.71 1.78 1.96 1.80 2.03 2.04 1.69 1.913.122 1.61 2.16 2.03 1.93 1.97 2.02 1.97 1.72 1.84 1.643.551 1.87 1.61 1.63 1.43 1.82 1.69 2.26 1.82 1.79 1.733.980 1.92 1.74 1.78 1.31 1.93 1.65 1.86 1.65 1.72 1.81

The nuclear and cortical stiffnesses (En and Ec) listed in units of 10�3 MPA

1967LABORATORY SCIENCE: CONTRIBUTIONS OF LENS GEOMETRY AND MECHANICAL PROPERTIES TO PRESBYOPIA

lens (45 GM1 and 45 GM2) restored 72% to 94% of theaccommodation of the 29-year-old lens. This repre-sents an improvement of 1.8 to 3.7 D over the original45-year-old lens. If the ciliary body displacement moreclosely follows that of the 29-year-old lens, simply re-shaping the lens will lead to an additional 3.0 D, whichis the lower limit for the prevention of reading glassessuggested by Weale13 (3.0 to 4.0 D). It appears that re-shaping the lens restores much of the accommodationloss. The 45 MM lens, however, provided only a smallimprovement in accommodative amplitude, yieldingan improvement of only 0.83 D over the original45-year-old lens.

These results suggest that themost significant lentic-ular change that can be made to improve accommoda-tion is to reshape the lens to a younger geometry. We

Figure 3. Optical power versus ciliary body displacement for 45-year-oldpower at varying nuclear stiffnesses and a 4.85 � 10�5 cortical stiffness. Rcortical stiffness.

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note that the idea that the geometry of the lens, ratherthan its mechanical properties, is the major factor incorrecting accommodation is not novel. It was hypoth-esized in 1995 by Pierscionek and Weale.33 Hence, ourstudy provides evidence for this hypothesis. Futurework will use an automated optimization routine tosearch for the optimal shape to obtain the largestaccommodative amplitude.

Sensitivity Analysis

Mechanical Properties In Table 3, the values with anasterisk were obtained using mechanical propertiessimilar to the range given by Heys et al.6 and thevalues with a dagger, to the range given by Burdet al.7 We find that the accommodative amplitude

geometry for varying nuclear and cortical stiffnesses. Left: Opticalight: Optical power at varying nuclear stiffnesses and a 3.98 � 10�3

OL 36, NOVEMBER 2010

Table 6. Polar thickness values.

Measurement 45 45 GM

Minimum 4.76 mm (37.5 year) 3.87 mm (25.5 year)Baseline 4.84 mm (45 year) 4.13 mm (29 year)Maximum 5.33 mm (83 year) 4.38 mm (29 year)

Minimum and maximum from Strenk et al.20 and baseline, from Burdet al.7

1968 LABORATORY SCIENCE: CONTRIBUTIONS OF LENS GEOMETRY AND MECHANICAL PROPERTIES TO PRESBYOPIA

values over the former range vary by 3.34 to 2.06 D, orapproximately 1.2 D, and over the latter range by 4.56to 4.29 D, or approximately 0.3 D. Note that we havebeen cautious in calculating these differences. Forinstance, one might conclude that the maximumchange in accommodative amplitude due to mechani-cal properties would be themaximum accommodativeamplitude (w6.9) minus the minimum accommoda-tive amplitude (w0.5) giving a change in accommoda-tive amplitude of approximately 6.4 D, which is muchlarger than that gained from lens reshaping. However,this would be an incorrect analysis as it would con-sider the difference between a lens with a 45-year-old nucleus and a 29-year-old cortex and a lens witha 29-year-old nucleus and a 45-year-old cortex, whichis not physically realistic. The most representativevalues are generally on the diagonal of Table 3, whereboth En and Ec increase from the upper left to the lowerright. The results thus obtained show that over a rangeof mechanical properties given in the literature for29-year-old to 45-year-old lenses, the maximumchange in accommodative amplitude due to variationsin mechanical properties appears to be approximately1.2 D. This is consistently lower than the increases inaccommodative amplitude obtained by reshapingthe lens, as shown in Tables 4 and 5, again suggestingthe change in geometry is the most significant factor inpresbyopia.

Tables 4 and 5 show that regardless of themechanicalproperty combination used, reshaping the 45-year-

Table 7. Increases in the accommodative amplitude (in diop-ters) when the 45-year-old lens is reshaped to the 29-year-oldlens geometry using the polar thickness values in Table 6.

To: / 45 GM1 45 GM2

From: Y Min Base Max Min Base Max

45 Minimum 3.45 3.74 3.7 1.75 1.85 1.9845 Baseline 3.41 3.7 3.65 1.71 1.81 1.9445 Maximum 3.17 3.46 3.41 1.47 1.57 1.7

GM1 Z maximum ciliary body displacement of 0.361 mm; GM2 Z maxi-mum ciliary body displacement of 0.277 mm

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old lens to the geometry of the 29-year-old lensresulted in an increase in accommodative amplitude,which was generally larger than the variations ob-served due to changes in the lens’mechanical proper-ties. Hence, the results of our sensitivity analysissupport the main thesis of this paper. Also note thatthe improvement in accommodative amplitude tendsto decrease with increasing Ec and En alongthe diagonals of Tables 4 and 5. However, thereare some deviations from this observed trend; forexample, in Table 4 when En and Ec increase from0.391 and 1.407 to 0.492 and 1.836, respectively, the im-provement in accommodative amplitude increasesslightly from 3.97 to 4.08. However, these fluctuationsin trend are again small comparedwith themagnitudeof the increases in accommodative amplitude them-selves. (The difference between 3.97 and 4.08 on the di-agonal of Table 4 is only a very small fraction of eitherof these values.) Hence these fluctuations again do notaffect ourmain conclusion regarding the efficacy of re-shaping. The interaction between the various lenscomponents and properties (nucleus, cortex, capsule,zonular fibers, gradient refractive index distribution,mechanical properties, etc) is rather complex, andmakes an interpretation of such fluctuations not onlydifficult, but also beyond the purpose of this investiga-tion. Hence, we do not elaborate on them. Finally,whether Weale’s13 criterion of 3 diopters’s worth ofimprovement was met depended on the exact mecha-nical properties and the maximum ciliary bodydisplacement.

In Figure 3, we observed that the optical power fora given displacement increased as the nuclear Youngmodulus increased, regardless of the value used forthe cortical Young modulus Ec (illustrative results

Figure 4. Refraction of light through the fully accommodated29-year-old lens with a polymer infill of lower index of refraction.

OL 36, NOVEMBER 2010

Figure 6.Optical power ranges andmaximumoptical powers versusrefractive index.

Figure 5.Optical power versus ciliary body displacement for variousinfill refractive indices.

1969LABORATORY SCIENCE: CONTRIBUTIONS OF LENS GEOMETRY AND MECHANICAL PROPERTIES TO PRESBYOPIA

are shown for 2 values of Ec). We also found that theresults for a low value of Ec (Figure 3, A) showed thereversal in accommodation mechanism observed byAbolmaali et al.,17 who tried inserting the much lowerYoung moduli (Ec Z 1.7 � 10�4 MPa and En Z 2.2 �10�4 MPa) into the Burd et al.7 finite element model,keeping all other mechanical properties the same asthose of Burd et al. As noted by Abolmaali et al.,17 inthis scenario, the optical power of the lens increasesslightly with increasing equatorial displacement andthen falls off.

We realize that this sensitivity analysis presentsonly a limited attempt at determining the impact ofvariations in the mechanical properties (eg, we havetested only the impact of the variation in Young mod-ulus of the cortex and nucleus). Further studies areneeded to determine the typical range of mechanicalproperties encountered in the actual presbyopic pop-ularion, and a more complete sensitivity analysis willhave to be performed to determine whether reshap-ing the lens would produce favorable results forevery patient. However, our preliminary resultsindicate that the geometry of the lens has a greaterimpact on its accommodative capacity than its exactmechanical properties. Our results therefore warrantfurther study in this area. Finally, the assumption ofa clear division between 3 homogeneous regionsdnucleus, cortex, and capsuledwill also have to bescrutinized to ensure the acceptable accuracy of thesesimulations. (For example, Burd et al.,7 although notexcluding the possibility of a nonhomogeneous lens,stated that there were insufficient grounds to con-clude from Fisher’s data that the nucleus stiffness

J CATARACT REFRACT SURG - V

was significantly different from the stiffness of thecortex.)

Geometric Properties As mentioned above, the resultsfor the first experiment of the geometry sensitivityanalysis are given in Table 7. For dmax Z 0.361 mm,the difference between the maximum and minimumchange in accommodation amplitude was 14% of thebaseline value used in the experiments above. Notethat thiswas the largestpossible changedue to the thick-ness variations modeled here. For dmax Z 0.277 mm,the difference between the maximum and minimumchange in accommodative amplitude was 28% of thebaseline used in the experiments above. In all in-stances, the change in improvement in the accommo-dative amplitude by lens reshaping was minimallyaffected by variations in lens thickness and the previ-ous conclusion remained valid. That is, if the ciliarybody displacement more closely followed that of the29-year-old lens, simply reshaping the lens wouldlead to an additional 3.0 D, which is the lower limit(3.0 to 4.0 D) for the prevention of reading glassessuggested by Weale.13

The effect of the geometry transformations in thesecond experiment of the geometry sensitivity studyon the lens curvature at the poles was generally as fol-lows: Scalings that thickened the lens increased its cur-vatures, and vice versa. The details of these studies arenot shown here as any number of similar studies couldbe done; however, the results were intriguing. Of allthe target geometries tried, the original 29-year-oldlens geometry specified by Burd et al.7 provided thegreatest improvement in accommodative amplitude

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over that of the 45-year-old lens geometry. This was byno means a complete study. (We continue to work onan optimization routine to more robustly determinethe optimum shape for the greatest increase in accom-modation amplitude.) However, it suggests that the29-year-old shape may be very close to the optimumtarget shape for maximizing the accommodativeamplitude.

Finally, while we examined the sensitivity to the tar-get geometry, the improvement in accommodativeamplitude due to reshaping also depends highly onthe geometry of the original lens. In this study, westarted with the 45-year-old lens given by Burdet al.,7 which is arguably younger and thinner thanthe lenses that would be considered for presbyopiacorrection in practice. If the initial lens geometry(here 45-year-old lens geometry) is more similar tothe final lens geometry after reshaping (here 29-year-old lens geometry), the amount of improvement inaccommodative amplitude due to that reshaping willdecrease. However, since the lens continues to growwith age, older lenses will generally be even fartherfrom the younger target geometry. Therefore, thepotential increase in accommodation amplitude dueto lens reshaping may in fact be even larger than thevalues obtained here. Future studies will be neededwith models covering a larger distribution of prepres-byopic and postpresbyopic lenses to more preciselyquantify the magnitude of the improvements.

Effect of Changing the Optical Properties

The results in Figures 5 and 6 show it is possible toincrease the accommodative amplitude by increasingthe index of refraction of the nucleus. Perhaps moreimportant, however, they also reveal that themaximum optical power is a stronger function of therefractive index of the infill material than the accom-modative amplitude. In other words, attempting toincrease the accommodative amplitude of a lens bythe infill method would cause an even greater increasein the maximum optical power, giving rise to the riskfor creating a myopic eye. We therefore consider thisintervention to be less effective than reshaping thelens to the 29-year-old geometry. However, the errorin maximum optical power could be changed witha subsequent refractive procedure such as laser insitu keratomileusis.

In conclusion, a finite-element analysis was used toinvestigate the mechanical deformation and resultingchange in focal length of the human crystalline lensduring accommodation. The mechanical modelsdrew upon the 29-year-old and 45-year-old geometriesandmechanical properties given in Burd et al.7 The op-tical properties of the lenseswere based on the gradient

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index of refraction distributions given by Blaker.11 Thefocal lengths (and resulting amplitude of accommoda-tion) were calculated with a ray-tracing algorithm de-veloped specifically for this application.

Our analysis suggests that of the 2 main lenticularchanges possibly associated with presbyopia(namely, changes in lens geometry and in mechanicalproperties), changes in lens geometry are the pre-dominant factor in the advent and progression ofpresbyopia. We found that simply reshaping the45-year-old lens to the geometry of the 29-year-oldlens restored its accommodative amplitude to 72%to 94% of that of the 29-year-old lens, depending onciliary body displacement. This is an improvementof 1.8 to 3.7 D over the 45-year-old lens. Conversely,restoring the mechanical properties led to a 0.7 Dimprovement. The upper range of this improvementoverlaps the range suggested by Weale13 requiredto restore the loss of accommodation of the presbyo-pic lens. Also, these improvements were based on45-year-old lens geometry. The improvement wouldprobably be larger when starting from the geometryof an older lens.

Several additional studies were conducted to deter-mine how sensitive the current baseline models wereto changes in mechanical properties, geometry, andoptical properties. It was found that the conclusionsfrom the baseline study remained valid acrossa wide distribution of elastic moduli and lens thick-nesses given in the literature. That is, changes in lensgeometry gave the largest changes in accommodativeamplitude. Trying to increase the accommodative am-plitude further after lens reshaping by changing the in-dex of refraction of the nucleus was not successful.

This study represents an initial investigation into thefundamental causes of presbyopia with the aim ofdeveloping a sufficient understanding of the mecha-nisms of presbyopia to develop methods for restoringthe accommodative amplitude. In this feasibilitystudy, we used the most complete models and lenticu-lar properties available. However, continual improve-ment of these models is required for enhancedunderstanding. In particular, future modeling effortswill include more detailed mechanical property distri-butions of prepresbyopic and postpresbyopic lenses;more detailed geometry distributions of the prepres-byopic and postpresbyopic lenses; further study ofthe system-level contributions and component-levelinteractions of the lens (ie, how accurately our discrete3-component system [nucleus, cortex, capsule] modelsthe lens); and more detail about the distribution ofciliary body displacements of prepresbyopic and post-presbyopic lenses, the distribution of zonule attach-ment locations in prepresbyopic and postpresbyopiclenses, and the distribution of the gradient index of

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refraction contours in prepresbyopic and postpres-byopic lenses.

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36, NOVEMBER 2010

First author:Dominique Van de Sompel

Department of Mechanical and Aero-space Engineering, Princeton Univer-sity, Princeton, New Jersey, USA